Even Symmetry

A function f (t) is said to possess even symmetry when

f (t) = f (-t)

In such a case the function is called even function. Geometrically, an even function is symmetrical not only with respect to the vertical axis passing through the origin t = 0 but also T with respect to all vertical lines at n T/2 (n= ± 1, ± 2, ....,) since all periodic functions with period T also satisfy the condition f (t) = f (t + T). Some of the waveforms possessing even symmetry are shown in Fig.

FIG. Even waveforms possessing symmetry

For even functions, f0 (t) = 0 and therefore from

αn = 2/T ∫0T/2 fe (t) cosnω0t dt

Since fe (t) and cos nω0t are even, the product fe (t) cos nω0t is also even. Therefore applying the condition given in condition.we obtain

αn = 4/T ∫0T/2 fe (t) cosnω0t dt

Similarly, from above

bn = 2/T ∫–T/2T/2 fe (t) cosnω0t dt

Since the product fe (t) sin nω0t is odd, the application of property of even functions yields.

bn = 0

Also, αn = 1/T ∫–T/2T/2 fe (t) dt

or αn = 1/T ∫–T/2T/2 fe (t) dt

Equation above shows that the coefficients of all sine terms are zero. Thus, the Fourier series of an even function contains only cosine terms and the constant term (if α0 ≠ 0). It is to be noted that for an even function, we need only integrate over half period, that is it is sufficient to integrate from 0 T/2 to - and then multiply the result by 2 in oder to evaluate αn.

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